## The Tracking LQG Controller

Previously considered LQG controllers were designed for vibration suppression purposes, where the commanding signal was zero. A more complex task includes a tracking controller, where a structure must follow a command. It requires tracking performance in addition to vibration suppression properties. This is the case of controllers for radar and microwave antennas, such as the NASA Deep Space Network antennas. This kind of controller should assure zero steady-state tracking error, which is achieved by adding an integral of the plant position to the plant statespace representation, as reported in [4], [36], [39], [42], [80], [118], and [142]. The closed-loop system configuration of the tracking LQG controller is shown in Fig. 11.5. In this figure (A,B, C) is the plant state-space triple, x is the state, x is the estimated state, xf is the estimated state of a flexible part, r is the command, u is the control input, y is the output, y is the estimated output, e = r - y is the servo error, ei is the integral of servo error, v is the process noise of intensity V, and the measurement noise w is of intensity W. Both v and w are uncorrelated: E(vwT ) = 0, V = E(vvT ), W = E(wwT ) = I, E(v) = 0, and E(w) = 0.

Figure 11.5. The tracking LQG controller with an integral upgrade.

Figure 11.5. The tracking LQG controller with an integral upgrade.

For the open-loop state-space representation (A, B, C) of a flexible structure the state vector x is divided into the tracking, xt, and flexible, Xf, parts, i.e., x =

The tracking part includes the structural position, and its integral, while the flexible mode part includes modes of deformation. For this division the system triple can be presented as follows (see [59]):

" a |
af ' |
' bt ' | |

, b = | |||

0 |
Af _ |
_ Bf _ |

The gain, Kc, the weight, Q, and solution of CARE, Sc, are divided similarly to x, k.

Sctf Scf

The tracking system is considered to be of low authority, if the flexible weights are much smaller than the tracking ones, i.e., such that QJ»¡Qf|. It was shown by Collins, Haddad, and Ying [20] that for qf = 0 one obtains scf = 0 and sctf = 0. This means that the gain of the tracking part, kct, does not depend on the flexible part. And, for the low-authority tracking system (with small qf), one obtains weak dependence of the tracking gains on the flexible weights, due to the continuity of the solution. Similar conclusions apply to the FARE equation (11.8).

This property can be validated by observation of the closed-loop transfer functions for different weights. Consider the transfer function of the Deep Space Network antenna, as in Fig. 11.6. Denote by in and 0n the identity and zero matrices of order n, then the magnitude of the closed-loop transfer function (azimuth angle to azimuth command) for qt = i2 and qf = 010 is shown as a solid line, for qt = i2 and qf = 5 x I10 as a dashed line, and for qt = 8 x I2 and qf = 010 asa dot-dashed line in Fig. 11.6. It follows from the plots that variations in qf changed the properties of the flexible subsystem only, while variations in qt changed the properties of both subsystems.

Note, however, that the larger Qf increases dependency of the gains on the flexible system; only quasi-independence in the final stage of controller design is observed, while separation in the initial stages of controller design is still strong. The design consists therefore of the initial choice of weights for the tracking subsystem, and determination of the controller gains of the flexible subsystem. It is followed by the adjustment of weights of the tracking subsystem, and a final tuning of the flexible weights, if necessary.

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